AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
“Theory Guides; Experiment Decides.”
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Experimental vibration analysis of engineering structures is a field
of increasing importance and popularity for researchers as
consequence both of
significant technological improvement of measurement
equipments and theoretical formulations
and of the extreme importance on the structural safety,
serviceability conditions and durability of vibrating structures.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Experimental vibration analysis
is one of the most important tools for analysing dynamic
properties of mechanical structures as the information obtained is
used in the development or modification of structures to obtain a
desired dynamic behaviour.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
This course is designed to use the experimental techniques in
vibration measurements and thus to provide the students
especially for the ones working on structural dynamics,
mechanical vibrations and modal testing areas by providing
unique inside on the general understanding of vibration test
planning, selection and use of exciters, transducers and sensors,
data collection, processing and assessment in particular with
hands on environment for modal analysis and testing.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Therefore, the course mainly focuses on
- investigating structural vibrations by putting particular
emphasize on the real application of experimental techniques in
vibration measurements by maintaining the balance between
theory and practical training.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Structures Lab Capabilities
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Structures Lab Capabilities SOFTWARE
MATLAB 2009a ANSYS 11.0 MSC PATRAN/ NASTRAN 2007r1 NI LabVIEW 8.6
HARDWARE
B&K 6 channel Pulse portable data acquisition unit with special softwareof FFT Analysis, Time Data Record, Modal Test Consultant,Operational Modal Analysis, Reflex Modal Analysis Software
B&K Modal Vibration Exciter (200N) B&K Impact Hammer
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Structures Lab Capabilities HARDWARE - Cont’
Various B&K Single-axis, Triaxial and miniature accelorometers,Empedance head.
Keyence Laser Displacement Sensor Polytec Scanning Laser Vibrometer
(New! – METUWIND Structural Dynamics LAB) Agilent Signal Generator Hameg Oscilloscope
Additionally, Various Uni-axial Strain Gauges and Installation Kits Dedicated equipment for smart structure applications comprising programmable
controller (SS10), high voltage power amplifiers, high voltage power supplies,preamplifiers and piezoelectric (PZT) patches in various size and shape.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Lecture # 1
1. Modal Analysis Theory
1.1 Theoretical Basis and Terminology
1.2 Modal Analysis of SDoF Dynamic Systems
1.3 Modal Analysis of MDoF Dynamic Systems
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Theoretical Basis and Terminology
Modal Analysis Theory
Theoretical (Analytical) Experimental
(Modal Testing)
Aim is to develop “Reliable Dynamic Models”!!
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Application of Modal Analysis:
• Identification and evaluation of vibration phenomena
• Validation, correction and updating of analytical dynamic models
• Development of experimentally based dynamic models
• Structural integrity assessment
• Structural modification and damage detection
• Reduction of mathematical models
• Determining, improving and optimising dynamic characteristics of
engineering structures.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Wind induced Vibration
Tacoma Narrows Bridge (Washington State), 1940
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Forced Vibration - Resonance
London Millenium Bridge Opening,2000
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Ground Resonance
A Chinook Helicopter
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Flutter
Piper PA-30 Twin Comanche Aircraft Tail Flutter Test,1966
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Flutter
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Assumptions:
• Structure is linear and time-invariant
• Structure obeys Maxwell’s Reciprocity Theorem.
• About FRFs (Frequency Response Functions)
• The positions of the shaker and accelerometer are reversed
in multiple single-input RECIPROCITY checks!
i.e. Various seperate single-input tests with the shaker located at different position for each test.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Assumptions:
• Structure obeys Maxwell’s Reciprocity Theorem.The measured FRF for a force at location j and response at location i should
correspond directly with the measured FRF for a force at location i and response
at location j. The FRF matrix is symmetric and this property can be used as a
check on the quality of the measured data.
MODAL ANALYSIS is the process of determining the
inherent dynamic characteristics of a system in the forms of
Natural Frequencies, Damping Factors and Mode Shapes.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
The three main phases of modal testing
• The theoretical basis of vibration,
• Accurate measurement of vibration (Controlled testing conditions),
• Realistic and detailed data analysis (Signal processing, Range of
curve fitting procedures in an attempt to find the mathematical model which
provides the closest description of the actually observed behaviour).
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Theoretical Route to Vibration Analysis
Description of
Structure
Vibration
Modes
Response
Level
Structure will vibrate under
given excitation condition
SPATIAL MODEL
• Mass
• Stiffness
• Damping
MODAL MODEL
• Natural Frequencies
• Mode Shapes
• Modal Damping Factors
RESPONSE MODEL
• Set of FRFs
• Impulse Responses
Structure's physical characteristics
Structure's behaviour as a set of vibration modes
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Experimental Route to Vibration Analysis
Response
Properties
Vibration
Modes
Structural
Model
Experimental Modal
Analysis
SPATIAL MODEL
• Mass
• Stiffness
• Damping
MODAL MODEL
• Natural Frequencies
• Mode Shapes
• Modal Damping Factors
RESPONSE MODEL
• Set of FRFs
• Impulse Responses
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Basic Vibration Theory
SDOF Systems
(Single Degree of Freedom)
MDOF Systems
(Multi Degree of Freedom)
Continuous Systems
(Infinitely many
number of DOF)
DOF: The minimum number of independent coordinates required to determine
completely the motion of all parts of the system at any instant of time.
A different selection of coordinates will lead to different equations of motion
but end up with same natural frequencies regardless of the choice of coordinates
(i.e. same system!)
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Basic Vibration Theory
Terms Vibration Type Description
External Excitation
• Free • Forced
• Vibration induced by initial inputs only• Vibration subjected to one or more continuousexternal inputs
Presence of Damping
• Undamped• Damped
• Vibration with no energy loss or dissipation• Vibration with energy loss
Linearity of Vibration
• Linear Vibration • Non-linear Vibration
• Vibration for which superposition principle holds• Vibration that violates superposition principle
Predictability • Deterministic • Random
• The value of vibration is known at any given time• The value of vibration is not known at any giventime but the statistical properties of vibration areknown
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
The Phasor
Phasor is a vector that rotates in a counterclockwise direction with Angular velocity in the complex plane.
A
Modal Analysis Theory
ω
{ }
{ } tjjtj
j
tj
eeAeAA
jSinCose
jwhereeAtjSintCosAA
ωφφω
θ
ω
θθ
ωω
⋅⋅=⋅=
±=
−=⋅=+=
+
±
1)(
11
)()(
1)()(
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
The Phasor
Modal Analysis Theory
Real
Imaginary
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
The Phasor
Modal Analysis Theory
)(2)(2
222
2
)( 2
πωω
ω
ωω
ωω
ω
ωωπ
+
+
=−=
=
==
tjtj
tj
tjtj
AeAe
Aejdt
Ad
AeAejdtAd
where
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Although very few practical structures could realistically be modeled by
SDOF System, a mode complex multi-degree-of-freedom (MDOF)
system can always be represented as the linear superposition of a
number of SDOF systems.
(a) Undamped
(b) Viscously Damped (c)
(c) Hysterically (or structurally) Damped (d or h)
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis of SDOF Dynamic Systems
Dynamic Properties of Mechanical Systems:
• Mass (Responsible for Inertia), • Stiffness (Responsible for Elastic Forces),• Damping (Responsible for Dissipative Forces)
(a)Undamped
Spatial Model: m, k (Simple Harmonic Oscillator = Spring&Mass System)
If no forcing;
(Modal Model))or (,)(
0
0 nti
mkxetx
kxxm
ωωω ==
=+0)( =tf
Modal Analysis Theory
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a) Undamped
Response Model
(In the form of FRF)
x and f are complex to accommodate both the Amplitude and Phase information
Complex if damping is not zero!Real if damping is zero!
)(1)(
)(
)(,)(
2
2
ωαω
ω
ω ωω
ωω
=−
==
=−
==
mkfxH
fexemk
xetxfetf
titi
titi
Force HarmonicResponsent Displaceme Harmonic)( =ωH
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a) Undamped
Circular Frequency (repetitiveness of the oscillation)
]/[11
][2][2
][]/[2
]/[2
]/[
scycleHz
Hzs
cyclecyclerad
sradf
srad
nnnnn
n
=
====πω
πω
πω
πω
ω
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a)Undamped
Simple Harmonic Motion:
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a)Undamped
Recall Phasor!
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Useful quantities describing the vibration;
Average Value:
Mean Square Value:Square of displacement is associated with a system’s potential energy.Average of the displecement squared is also a useful vibration property.
Root Mean Square (RMS) Value:Square root of the Mean Square value is commonly used in specifying vibration.
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping (c)
Viscous dashpot c or damper, physical model for dissipating energy
Equation of motion for free vibration case;
In Laplace Domain,
]/[],/[],/[],[0
skgmNscmNkkgmkxxcxm
→→→=++
mk
mc
mcs −
±−=
2
2,1 22
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping
Overdamped: Both roots are real.
Underdamped: Two roots are complex conjugate.
Critically damped: Two equal real roots.
Condition Damped Criticallyfor Constant DampingConstant Damping Ratio Damping
frequency natural Undamped
222tCoefficien Damping Critical
=→
→
===→
ξ
ω
ω
n
nc mmkmkmc
mk
mc
⟩
2
2
mk
mc
⟨
2
2
mk
mc
=
2
2
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
frequency natural Damped
1s
quantity essDimensionl
21,2
→
−±−=
→=
d
d
nn
ccc
ωω
ξωξω
ξ
(b) Viscous Damping
Overdamped:
Underdamped:
Critically damped:
Damping ratio for critically damped systems seperates oscillatory motion from nonoscillatory motion and critical damping is the value of damping that provides the fastest return to zero without oscillation.
1⟩ξ
1⟨ξ
1=ξ
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Critical Damping
Army gun firing – Explosion and recoil take a few miliseconds With recovery of less than 1 second.
In SDOF systems, it all about using all the spring’s potential energy!
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(c) Structural (Hysteretic) Damping (d or h)
Describes more closely the energy dissipation mechanism.
By making viscous damping rate vary inversly with the frequency.
Provides much simpler analysis for MDOF systems!
ωdc =
ξηω
ηω
ηξ
ωη
ωω 21222
Factor Loss Damping Structural
at n = →====
==→
=
mk
kmk
kmc
cc
kd
kc
c
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
A common unit of measurement for vibration amplitudes and RMS values isthe decibel (dB).
The decibel is defined in terms of the base 10 logarithm of the power ratio oftwo electrical signals (or as the ratio of the square of the amplitudes of twosignals)
Voltage ratios in dB are calculated by
dB Scale expands of compresses vibration response information
→
=
→
2
110
2
110
2
2
110
log20dB
log20log10dB
VV
xx
xx
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Decade: A 1:10 increase or decrease of a variable, usually in frequency.
Example: A 20 dB/decade gain: A gain change of 20 dB for each 10 foldincrease or decrease in frequency.
Numerically:( )
( )( ) decadedB
decadedB
sradsrad
dBsrad
/40100log20For /01log20For
/1001010 and/11010
2010log20/10
102
101
10
=⇒=⇒
=×=
==
ωω
ω
20 dB decrease!
20 dB increase!
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Octave: A doubling or halving, usually applied to frequency.
Example: A 6 dB/octave gain: A gain change of 6 dB for each doubling orhalving of frequency.
Numerically:( )
( )( ) octavedB
octavedB
sradsrad
dBsrad
/2620log20For /145log20For
/20210 and/52
10
2010log20/10
102
101
10
≅⇒≅⇒
=×=
==
ωω
ω
6 dB decrease!
6 dB increase!
Slopes can be defined as either dB/octave or dB/decade. octavedBdecadedB /6/20 ≅∴
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis of SDOF Dynamic Systems
Definitions of FRFs;
Modal Analysis Theory
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
A most effective way of investigation for modal analysis is using the Frequency Response Function (FRF)
RECEPTANCE:(recall)
)(fx
fexe)(H
fexe)mk(
xe)t(x,fe)t(f
ti
ti
titi
titi
ωαω
ω
ω
ω
ωω
ωω
===
=−
==
2
Force HarmonicResponsent Displaceme Harmonic)( =ωH
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Alternative forms of FRF;
MOBILITY:
2 : Phase
)()( :Magnitude
)()(
)(
)(,)(
Y
2
πθθ
ωαωω
ωωαωω
ω
α
ω
ω
ωω
ωω
+=
=
===
=−
==
Y
ifexei
fxY
fexemk
xetxfetf
ti
ti
titi
titi
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis of SDOF Dynamic Systems
Summary;
The reciprocals of the three FRFs of an SDOF system.
πθπθθ
ωαωωωω
α +=+=
==
2 : Phase
:Magnitude
A
2
Y
)()(Y)(A
)(A
)(Y
)(
ω
ω
ωα
1 Responseon Accelerati
ForceMassApparent
1 ResponseVelocity
ForceInpedance Mechanical
1 Responsent Displaceme
ForceStiffness Dynamic
==
==
==
Modal Analysis Theory
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping, c
Response Model (In the form of FRF)
( )
( )
( ) )c(imk)(F)(X)(A
)c(imki
)(F)(X)(Y
)c(imk)(F)(X)()(H
ωωω
ωωω
ωωω
ωωω
ωωωωωαω
+−−
==
+−==
+−===
2
2
2
2
:eAcceleranc
:Mobility
1 :Receptance
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(c) Structural (Hysteretic) Damping, d or h
Response Model (In the form of FRF)
( )
( )
( ) ihmk)(F)(X)(A
ihmki
)(F)(X)(Y
ihmk)(F)(X)()(H
+−−
==
+−==
+−===
2
2
2
2
:eAcceleranc
:Mobility
1 :Receptance
ωω
ωωω
ωω
ωωω
ωωωωαω
AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
1. Modal Analysis Theory
1.1 Theoretical Basis and Terminology
1.2 Modal Analysis of SDoF Dynamic Systems (cont’)
1.3 Modal Analysis of MDoF Dynamic Systems
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